Primary Resonance in a Weakly Forced Oscillator With Both Parametric Damping and Stiffness

We study the primary resonance of a parametrically damped Mathieu equation with direct excitation. Potential applications include wind-turbine blade vibration with cyclic stiffening and aeroelastic effects, which may induce parametric damping, and devices with designed cyclic damping for resonance manipulation. The parametric stiffness, parametric damping, and the direct forcing all have the same excitation frequency, with phase parameters between these excitation sources. The parametric amplification at primary resonance is examined by applying the second-order method of multiple scales. With parametric stiffness and direct excitation, it is known that there is a primary parametric resonance that is an amplifier under most excitation phases, but can be a slight suppressor in a small range of phases. The parametric damping is shown to interact with the parametric stiffness to further amplify, or suppress, the resonance amplitude relative to the resonance under parametric stiffness. The effect of parametric damping without parametric stiffness is to shift the resonant frequency slightly, while inducing less significant resonance amplification. The phase of the parametric damping excitation, relative to the parametric stiffness, has a strong influence on the amplification or suppression characteristics. There are optimal phases of both the direct excitation and the parametric damping for amplifying or suppressing the resonance. The effect of the strength of parametric damping is also studied. Numerical simulations validate the perturbation analysis.